Do I have to register the domain name?
If I'm using a website builder for instance and the domain name I want is already paid for do I still have to register it?
If the domain name is already paid for and the registrant is under your name, then it is yours.
There is no need for you to re-register the domain, in fact, you / anyone can't re-register the domain.
You can check it out by go to this url: http://www.whois.com/whois/YOUR_DOMAIN_NAME
*Replace "YOUR_DOMAIN_NAME", with actual domain name.
Hopefully that helps,
Web Design Melbourne
As Aditya said, there's no need ot register if a custom domain name is included in your subscription to a website builder (such as squarespace or wordpress). So it shouldn't be an issue.